Problem 15 - College Algebra with Modeling and Visualization

Question

Find all positive integers \(n\) for which the given statement is not true. $$ 3^{n}>6 n $$

Step-by-Step Solution

Answer

The positive integers for which the statement is not true are \( n = 1 \) and \( n = 2 \).

Step 1: Understand the Problem

We need to find positive integer values of \( n \) for which the inequality \( 3^n > 6n \) is **not** true. This means we are looking for values of \( n \) where \( 3^n \leq 6n \).

Step 2: Test Small Values

Begin by testing small integer values of \( n \) to see where \( 3^n \leq 6n \). Start with \( n = 1, 2, 3, \ldots \). For \( n = 1 \), \( 3^1 = 3 \) and \( 6 \times 1 = 6 \), so \( 3 \leq 6 \) is true. For \( n = 2 \), \( 3^2 = 9 \) and \( 6 \times 2 = 12 \), so \( 9 \leq 12 \) is true. For \( n = 3 \), \( 3^3 = 27 \) and \( 6 \times 3 = 18 \), so \( 27 > 18 \) is false.

Step 3: Determine Changing Point

Observe that starting from \( n = 3 \), the inequality \( 3^n > 6n \) holds true. This suggests finding a general turning point or deriving inequality behavior as \( n \) increases.

Step 4: Compare Growth Rates

Compare the exponential growth of \( 3^n \) with the linear growth of \( 6n \). The function \( 3^n \) grows exponentially, and \( 6n \) grows linearly. By testing further values, we see that \( 3^n \) continues to be greater than \( 6n \) for \( n \geq 3 \).

Step 5: Identify Solutions

Since we have determined that \( 3^n \leq 6n \) only holds for \( n = 1 \) and \( n = 2 \), these are the positive integers for which the original statement is not true.

Key Concepts

Exponential Growth

Exponential growth is a concept where a quantity increases by a constant factor over equal time intervals. This means that as time goes on, the quantity grows at a rate that becomes increasingly rapid. In mathematics, exponential growth is often represented using powers, such as \( a^n \), where \( a \) is the base and \( n \) is the exponent.

In the context of our exercise, the expression \( 3^n \) demonstrates exponential growth. As \( n \) increases, the value of \( 3^n \) grows very quickly compared to \( n \). For example, \( 3^3 = 27 \) grows much faster than \( 3^2 = 9 \), illustrating the rapid increase in magnitude.

Exponential growth is common in real-world situations like population growth, certain financial investments, and the spread of infectious diseases. Understanding its nature helps in comparing it with other growth types, like linear growth.

Linear Functions

Linear functions represent relationships where the increase or decrease between the values is constant. This is represented by a straight line on a graph. A linear function typically takes the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

In the inequality \( 3^n > 6n \), the term \( 6n \) represents a linear growth. Here, \( n \) is multiplied by a constant value of 6, indicating that the growth is steady and consistent as \( n \) increases. This means that every increase in \( n \) results in a proportional increase in the value of \( 6n \).

Linear functions are useful in situations that require consistent, predictable change over time, like predicting expenses or calculating speed. Recognizing the difference between linear and exponential growth is crucial in problem-solving and predicting behaviors in mathematical models.

Problem Solving in Mathematics

Problem-solving in mathematics involves understanding the problem, devising a strategy, and executing steps to find solutions. It is about applying mathematical concepts to interpret problems and find answers.

In our exercise, the problem required identifying the integer values of \( n \) where the inequality \( 3^n \leq 6n \) holds true. The strategy involved:

• Understanding the behavior of both exponential and linear functions in the inequality.
• Testing small values of \( n \) to observe where the inequality changes.
• Analyzing and comparing the growth rates of \( 3^n \) and \( 6n \).

This approach highlighted that for \( n = 1 \) and \( n = 2 \), the inequality was true. Beyond these values, the exponential growth surpassed the linear growth, offering a clear picture of the problem's nature.

Successful mathematics problem-solving also involves critical thinking, perseverance, and the ability to connect different mathematical concepts. Developing these skills helps students face increasingly complex problems with confidence.